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I would like to know a good way of sketching the surreal number line. I wonder if there is a format that is widely used.

In this video, Conway makes a quick drawing of a line, just like the real line (without gaps), and separates the integers to the infinities with some visual space, but no particular notation. However, I started wondering what would be a good way to draw the infinitesimals in the line. I do not think it is possible to do the same thing he did for infinities in this case, because it may give the impression that a real number has a "next" number.

For example:

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___ 0_$\ldots\epsilon\ldots$_? ____1 _2 _3 ___$\ldots$ __$\omega-1$ _$\omega$ ___

I wonder if representing the infinitesimals, or gaps (like in page 37 ONAG2) for that matter, in a precise manner is at all possible. Would a skecth like the one found above, put together with an explanation of its limitations, be a valid representation?

MTLaurentys
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    "All models are wrong, but some are useful." George E. P. Box. Your sketch has substantial limitations, but it can help thinking about the numbers. Usually one would show more detail, like the fact that there are even smaller infinitesimals which you can represent as $\epsilon^2$ and show some more transfinite numbers above $\omega$. It is also good to show $\frac \omega 2$ because many uses of $\omega$ do not allow cutting it in half. You have hinted at it with $\omega-1$. – Ross Millikan Oct 04 '20 at 19:31
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    Since you have ONAG, can you explain what deficiencies you find with the tree and line segments in the Frontispiece of ONAG? – Mark S. Oct 04 '20 at 19:36
  • @MarkS. I am very far from being an expert on the subject, so I might not have a proper deficiency, in the sense you are looking for. I see that everything is there, numbers and some gaps, but I could not merge all the line segments and (...)gaps together in a single line. – MTLaurentys Oct 04 '20 at 19:54
  • The video I linked got me thinking there was a standard way of writing in this form, because it was a single continuous line, but he probably just wanted to be brief. I should have been more critical before asking this. – MTLaurentys Oct 04 '20 at 20:02
  • A drawing won't distinguish the rational and the standard reals. – badjohn Oct 04 '20 at 20:20
  • "I could not merge all the line segments and (...)gaps together in a single line" If you connect the number line part of the frontispiece with a horizontal line, can you explain what more would you want it to do or suggest about the surreals that it fails to? – Mark S. Oct 04 '20 at 20:41
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    @MarkS. If I were to connect the segments, wouldn't it be something like: 0-(1/On)-...$\epsilon$...-...1_2_3...-(infinity)-...$\omega$...-On. It does not give the idea, to my weak understanding, that such gaps and infinitesilmals are... everywhere – MTLaurentys Oct 04 '20 at 21:00
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    Is there a picture of the real line that clearly gives you the idea that rationals and irrationals are everywhere? If so, perhaps we could emulate that to give a more satisfying picture of the surreals&gaps. If not, perhaps what you're looking for is impossible? – Mark S. Oct 04 '20 at 21:02
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    The real number line gives me this idea - every segment and point is composed of such numbers. I worry this may be contradictory with what I state about the surreal number, and it may as well be. When I draw a contiguous number line, I immediately think about the reals and I am starting to believe this may be the problem. The trouble I had may be because this question is mixed with some remaining misunderstanding I still have about the continuum the reals form. Hmmm, I have to study more. – MTLaurentys Oct 04 '20 at 21:12

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There are more surreal numbers than real ones, so they can't each get their own point on a line. In fact, the surreal numbers comprise a proper class, i.e. no set, be it of the points on a line or otherwise, is as big as the class of surreal numbers.

J.G.
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    I think of the real line as a way of drawing the reals, or perhaps a way of drawing $\mathbb{R}\times{0}$ inside of $\mathbb{R}^2$. I see little reason to restrict the use of a line to continuum-sized objects.

    For example, when visualizing Robinson's hyperreals, a line like the real line can be useful and is indeed used in Keisler's "Elementary Calculus: An Infinitesimal Approach". So I see no strong reason not to allow a "surreal line" picture.

     – Mark S. Oct 04 '20 at 20:40
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    @MarkS. It depends on what kind of representation you seek. If you want each SN to get its own point, you'll be disappointed. If you just want an analogy for how SNs are ordered, their topology etc., you might be OK. – J.G. Oct 04 '20 at 20:48
  • I guess what I'm saying is that it seems you assume that a "point" (drawn in a picture?) is a member of a set the same size as the reals, and I feel that's kind of arbitrary. I can't assign each real in an interval to a molecule of ink on a page, either. – Mark S. Oct 04 '20 at 20:52